How to prove that $\displaystyle \lim_{x\to2}{x^2}=4$ by using the formal definition of limit?
I tried to solve it, but the problem was to find delta that depends on epsilon.
Consider:
$\displaystyle \left |{x^2-4}\right |< \epsilon \Leftrightarrow \left |{x-2}\right |\left |{x+2}\right |< \epsilon$
For choosing $\displaystyle \delta$ take into account that $\displaystyle |x+2|$ is (for example) bounded on $\displaystyle [1,3]$.
Regards.
Fernando Revilla
On $\displaystyle (1,3)$ we have $\displaystyle |x+2|<5$,
then, on $\displaystyle (1,3)$
$\displaystyle |x^2-4|=|x-2||x+2|<5|x-2|<\epsilon \Leftrightarrow |x-2|<\epsilon/5$
Now, choose
$\displaystyle \delta=\min\{1,\epsilon/5\}$
Regards.
http://www.fernandorevilla.es/